On the design of membranes with increasing fundamental frequency

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By means of a relaxation approach, we study the shape design of a stiff inclusion with given area in a membrane in order to maximize its fun- damental frequency. As an eigenvalue control problem, the fundamental frequency is a concave function of the control, which is not described by the membrane shape, but by an element in a function space. First order optimality conditions allow to describe the optimal shape by means of a free boundary value problem.